Exponentially Weighted Moving Average Filter

The moving average filter regards each data point in the data window to be equally
important when calculating the average (filtered) value. In dynamic systems, however, the
most current values tend to reflect better the state of the process. A filter that places
more emphasis on the most recent data would therefore be more useful. Such a filter can be
designed by following the procedure used in developing the moving average filter. As
before, the starting point is the mean expressed as:

But in this case, consider also the mean
with one additional point

Since, therefore,

By shifting the time index back one
time-step, we obtain the corresponding expression for as:

To simplify the notation, let , which implies that . We can write the filter as:

This expression is known as the Exponentially
Weighted Moving Average Filter. When used as a filter, the value of is again taken as the filtered value of . Notice that now, calculation of does not require storage of past values of x,
and that only 1 addition, 1 subtraction, and 2 multiplication operations are required.

The value of the filter
constant, , dictates the degree of
filtering, i.e. how strong the filtering action will be. Since , this means that . When a large number of
points are being considered, , and . This means that the degree of
filtering is so great that the measurement does not play a part in the calculation of the
average! On the other extreme, if , then which means that virtually no filtering is being
performed.

The Exponentially Weighted Moving Average
filter places more importance to more recent data by discounting older data in an
exponential manner (hence the name). This characteristic can be illustrated simply by
describing the current average value in terms of past data. For example, since

,

then

Therefore,

i.e.

But ,

Therefore,

If we keep on expanding terms on the right hand side,
we will see that the contribution of older values of are weighted by increasing powers of . Since is less than 1, the
contribution of older values of becomes progressively smaller. The weighting on may be represented graphically by the following
plot:

Figure 2. Exponential weighting effect

What this means is that in calculating the
filtered value, more emphasis is given to more recent measurements.

The Exponentially Weighted Moving Average
filter is arguably the most commonly used noise reduction algorithm in the process
industries. However, it is known commonly by a another name; one that has its roots
in electrical circuitry that are used to produce smooth electrical signals.